There is something about Natsume's relationship with Reiko, or the lack of it.
Natsume cares about Reiko. He never met her. She's his age. She was dead before he was born. She is one of the most prominent characters in the series, second to only Natsume himself. Her choices drive the plot. We know next to nothing about her.
Nearly every yokai Natsume meets was deeply impacted by Reiko. She helped me, she hurt me, she held my name (my soul) in her hands and left it to rot. They are still waiting for her. In valleys and forests and bus stops.
The Book of Friends is the result of one of Reiko's most painful experiences. The physical manifestation of her choice to keep people at arm's length, to push people away before they could do it to her.
Even after settling with the Fujiwaras, Natsume felt so adrift. At the beginning of the series, he may have actually agreed with Reiko's philosophy. "It's better to be alone than to risk getting hurt." His trauma left him disconnected from everybody. The Fujiwaras, his classmates, the yokai. Natsume had no past he could reconcile, no future he could imagine, and even his present he walked through like a dream; doing everything in his power to leave no trace.
The book forced him to make connections. More than that it forced him to reflect on his own relationships. As Natsume continues to return names, he understands Reiko and her lonliness more and more... And he realizes that she was wrong.
Through the Book of Friends, Reiko tethers Natsume to his past, grounds him in his present. Makes him reach out to others in ways she couldn't. Allows him to imagine a future so much brighter than he could have envisioned for himself, even just a few years earlier.
But while Natsume has grown, Reiko has remained stagnant. This girl who only exists in memories of decades past, this woman who died before he was born.
So of course Natsume is trying to find out what happened to her. Hoping that somehow, Reiko found something for herself too. That at least for a while, she was happy.